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To solve the equation \(\cos(xy) = 3x + 9y\) using implicit differentiation, we differentiate both sides of the equation with respect to \(x\). Since \(y\) is a function of \(x\), we apply the chain rule to handle the product of \(x\) and \(y\) inside the cosine function.

### Step-by-Step Solution

Given:
\[
\cos(xy) = 3x + 9y
\]

1. **Differentiate both sides with respect to \(x\):**

\[
\frac{d}{dx}[\cos(xy)] = \frac{d}{dx}[3x + 9y]
\]

2. **Apply the chain rule to the left side:**
   - The derivative of \(\cos(u)\) is \(-\sin(u)\), where \(u = xy\).
   - Now, apply the product rule to differentiate \(xy\) with respect to \(x\).
   
\[
-\sin(xy) \cdot \frac{d}{dx}(xy) = 3 + 9 \frac{dy}{dx}
\]

3. **Differentiate \(xy\) using the product rule:**
   \[
   \frac{d}{dx}(xy) = x \frac{dy}{dx} + y
   \]
   Therefore, the left side becomes:
   \[
   -\sin(xy) \cdot (x \frac{dy}{dx} + y) = 3 + 9 \frac{dy}{dx}
   \]

4. **Simplify the equation:**
   \[
   -\sin(xy) \cdot (x \frac{dy}{dx} + y) = 3 + 9 \frac{dy}{dx}
   \]

5. **Expand both sides:**
   \[
   -\sin(xy) \cdot x \frac{dy}{dx} - \sin(xy) \cdot y = 3 + 9 \frac{dy}{dx}
   \]

6. **Group the terms involving \(\frac{dy}{dx}\):**
   \[
   -\sin(xy) \cdot x \frac{dy}{dx} - 9 \frac{dy}{dx} = 3 + \sin(xy) \cdot y
   \]

7. **Factor out \(\frac{dy}{dx}\):**
   \[
   \frac{dy}{dx}(-\sin(xy) \cdot x - 9) = 3 + \sin(xy) \cdot y
   \]

8. **Solve for \(\frac{dy}{dx}\):**
   \[
   \frac{dy}{dx} = \frac{3 + \sin(xy) \cdot y}{-\sin(xy) \cdot x - 9}
   \]

### Final Answer:
\[
\frac{dy}{dx} = \frac{3 + \sin(xy) \cdot y}{-\sin(xy) \cdot x - 9}
\]

Would you like further details on any part of the solution?

Here are some related questions for practice:
1. What is implicit differentiation, and how does it differ from explicit differentiation?
2. How is the product rule applied in implicit differentiation?
3. What is the chain rule, and why is it necessary in this problem?
4. Can you solve the same problem for different trigonometric functions, like \(\sin(xy)\)?
5. How would the solution change if the equation had a higher-degree polynomial on the right-hand side?

**Tip:** Always remember to apply the chain rule whenever differentiating a function of both \(x\) and \(y\) in implicit differentiation.

Use implicit differentiation to find dy/dx for cos(xy) = 3x + 9y.

Learn how to apply implicit differentiation to find dy/dx for the equation cos(xy) = 3x + 9y. This problem involves using the product rule, chain rule, and trigonometric derivatives. Suitable for students in advanced calculus courses.

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